Friday, October 29, 2010

Do You Believe in Algebra?

(Cross-posted on The Huffington Post).

Let me clarify. I'm not asking do you believe in Algebra in the same sense as do you believe in the tooth fairy (full disclosure: I do not). I'll posit that Algebra exists. Rather I'm asking if you believe in Algebra as a separate course/curriculum that we should teach in high school.

After my last post, Dean Shareski wrote a thought-provoking post that asked whether it was possible to offer a customized educational experience in a standards-based system of education.

Our current system and structure fights personalized learning with nearly every new policy and protocol it can generate. The system craves standardization while we desperately need customization. These competing ideals butt heads constantly and for those teachers who do believe in personalizing learning, they live in perpetual frustration. . . In the end, without a restructuring of time and current curriculum requirements the best we can hope for is small pockets of success or the .02 percent of students whose passion happens to be trigonometry or Shakespeare.

Dean later acknowledges, however, that while he wants personalization, he also wants students exposed to a broader range of ideas:

While I'm busy advocating for changes that might support an education that fuels and fosters students' passions, I worry that we lose sight of what a liberal education is all about. They don't know what they don't know. Providing students with broad experiences that invites them to develop a variety of skills, understand and appreciate diverse perspectives and potentially uncover hidden talents and interests speaks to a fairly well accepted purpose of school. . . If we were truly starting education from scratch today, I can't imagine we'd build the same system we have. There would be lots of discussion as to what types of content all students need. Even if core content and skills could be determined, we'd never teach them all as segmented subjects taught in isolation in 45-minute increments.

And therein lies the dilemma - is it possible to provide in a systemic way a customized educational experience for all students that both allows and encourages them to pursue their passions, but also exposes them to the wide range of human endeavors that they may have little or no knowledge about and therefore wouldn't be able to even know if they were passionate about in the first place?

Which brings us back to Algebra. I teach in Colorado, which recently adopted the Common Core State Standards. In general, I believe the Common Core Math Standards (pdf) are much better than most standards that came before them. First, there are fewer of them, with 156 standards for grades 9-12. In addition, 38 of those standards are identified as "advanced" standards, which leaves us with 118 standards for all students spread out over four years of high school, or just under 30 per year. That's much, much more doable then what we had before, and I believe targets much more of what I would consider mathematics that is essential for people to know.

But it still begs the question of whether all students need these 118 standards. For example, do you believe that all students (scratch, that, all people) need to know "there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real?" (CCSS, N-CN 1). Or how about "prove the Pythagorean identity sin2(x) + cos2(x) = 1 and use it to find sin(x), cos (x), or tan(x) and the quadrant of the angle?" (CCSS, F-TF 8).

(My not-so-modest proposal is that no state legislature is allowed to require standards that they couldn't demonstrate proficiency on themselves. Since they are clearly successful adults and they are saying that these standards are necessary for all students to be successful, surely they'd be able to demonstrate proficiency by taking the same tests our students do. But I digress.)

As G.V. Ramanathan recently asked in the Washington Post:
How much math do we really need?
In an age of information abundance, when Wolfram Alpha can do pretty much all of high school math quickly and at no charge, do all students need to be able to know all 118 standards? When instructional videos (either homegrown or created by others like Khan Academy) exist that replicate many aspects of a traditional math classroom and allow students to learn the skills at a time and a place of their own choosing, what activities should be taking place in our math classrooms?

Consider these statistics:

1985: 3,800,000 Kindergarten students
1998: 2,810,000 High school graduates
1998: 1,843,000 College freshman
2002: 1,292,000 College graduates (34%)
2002: 150,000 STEM majors (3.9%)
2006: 1,200 PhD's in mathematics (0.03%)

(source: presentation by Steve Leinwand, American Institutes for Research at NCTM Regional Conference in Denver on October 7, 2010. His source U.S. Statistical Abstract)

There's lots we could talk about with those statistics, but I'm just going to focus on what percentage of our students truly need the Common Core Math Standards. I would suggest that it's most likely somewhere between the 3.9% and the 34%, which makes me wonder how "core" they really are. While I think Common Core, combined with replacing Calculus with probability and statistics as the capstone to high school mathematics for most students, would be an improvement on much of what we're currently doing, I'm still not sure whether teaching Algebra as a separate course is the best way to accomplish it - even for that small subset of our student population that is passionate about math and science.

Can we find a way to have students whose passion is math and science explore rich, meaningful mathematics that isn't divided up into courses (Algebra), semesters (first semester linear, second semester non-linear), and units (Chapter 5: Writing Linear Equations)? Can we do this in a meaningful way for all students, even those who currently don't have a passion for math and science? Can we do it in a mathematically coherent way that doesn't impact a student's ability to progress to higher-level mathematical thinking should they choose to do that? Can we do this within a system that - at its heart - is an assembly-line model designed to mass produce a fairly standard product?

I think this is essentially what Dean - and many of us - are asking ourselves. Is there a way to combine the best of both? The best of passion-based learning and a liberal arts education that exposes students to some "standard" body of knowledge that we believe all people should be exposed to. Can the current system - with all its flaws and all its successes - adapt to a personalized, on-demand, anytime, anywhere learning environment? Or do we have to start over with a system that is designed to meet the needs of the learner and one that - at its heart - is antithetical to a standards-based system?

I honestly don't know. Because while I do believe the current system is designed to meet the needs of a rather small portion of our students, I'm not sure I can clearly define what mathematics education would look like in such a new system. As I stated in a previous post, I believe we can have high standards without standardization, yet like Dean I struggle to envision exactly what that looks like in practice in any kind of systemic way.

So, do you believe in Algebra as a separate course/body of study in high school? Or, like the tooth fairy, is Algebra - and standards-based, one-size-fits-all education - something we should've outgrown by now?


  1. I think the content is not the issue--it's the delivery. In fact, understanding algebra is very important, even in a liberal arts education. However, like the thinking that drives our standards, learning in math is "delivered" in a linear way (seems logical on one hand). Math concepts are all around us and really beautiful, but generally taught in a vacuum of empty equations without meaning and context.

    I would consider myself a math failure with a bulging right brain, however I think my path was determined precisely when I was introduced to Algebra. I excelled in all things verbal, historical and artistic, mostly because math did not have any meaning for me. However, when, as an adult, I learned about fractal geometry and math as art, I felt (and still feel) robbed. If someone had shown me math in its visual and relative form, and in the context of nature, instead of meaningless numbers and letters, I may be getting a Ph.D. in math as we speak. But because of my not-so-stellar math experience, I'm getting a Ph.D. in STEM education in an effort to find new ways to bring relevance to math and science for a larger number of students with visual learning tools. The fact that there is such a low number of students in higher math is not a foundation for an argument for its irrelevance, rather a reflection of the poor job our schools do in engaging students in a topic of relevance and beauty.

  2. exactly our focus of research for like 3 years.. dang - i'll try to be brief.

    we're thinking the only skill learners need today is how to learn. knowing what to do when they don't know what to do.

    we're thinking if we focus on and master the process, the topic doesn't matter. so personalization can happen - choice can happen - learning can be per passion. this creates the perfect setting for Coyle's deep practice. changing passion later is fine because the process has become 2nd nature. self-construction, self-automation, but most important, ongoing hunger for learning results.

    the benefits:
    we can prune out what a learner doesn't need - making learning more humane, and
    we can amp what he/she does need - making the learner more remarkable.

    we're attempting to experiment and explain it all here:

    to your specific question of a lover of math.. i suggest we trust mathematical thinking (to me the only basic)to happen naturally in every learning endeavor. because it will. and that's all the masses need. and right now - it's way more than the masses are getting. i'm afraid we've created math haters by pushing so many bits and pieces... reducing this beautiful language to some abstract grammatical structure.

  3. As a 20+ year software engineer, I'm one of the STEM 3.9%, with a full slate of Calc and engineering math in college. Now that I am a preservice teacher readying myself for teaching middle schoolers a semester of Algebra, I am reflecting on what math I have used in my career and life, how I have used it, and how I have learned it.

    It turns out that very little of the interesting math I have used came directly out of schooling. The statistics skills of my youth (born from a love of baseball) I learned on my own before I ever saw that subject in school. The math that has stuck with me and colors the way that I view the world has been primarily learned "on the job" - connected with an immediate need or interest and often self-taught.

    So, to build on Jennifer's comment and to address Dean Shareski's concern for exposing students to a broader range of ideas (which I share), I think the focus should be on helping students experience the world through a numerical lens, and letting them find their OWN questions. Once they are personally invested in an inquiry, learning the mathematical tools they need will come more naturally. It's not necessary that the question be a "practical" application of math, just that it's born of genuine curiosity.

    In a Research in Practice post Ben Blum-Smith wrote about Avital Oliver's opinion on teaching math, and these two sentences are continually in my mind when thinking about math: "To Avital, it’s not mathematics unless you’re trying to answer a question you already had. You’re not teaching mathematics unless you’re working with people to build tools to find answers they already wanted." How much of school math fits that description?

  4. ...I think the focus should be on helping students experience the world through a numerical lens, and letting them find their OWN questions....

    Nice! I wanted to chime in on this notion and say that it is exactly this kind of empowerment that we're developing with The concept is for everyone to use an economic platform that ensures that everyone within can see the economy through the eyes of an unerring econometrician, "See the truth first, THEN make your decisions involving risk." Banks and politicians don't appear to like the idea very much, yet, but it is expected to have this effect for a while. Since everyone must have the right to perfectly understand exactly what's happening within their economy (by having the math already done), people won't be able to use the "lack of access to information" as an excuse to justify their importance, or brush off responsibility for the decisions they've made, e.g. bad investment decisions, or poor policy decisions.

  5. Jennifer - While I agree that the delivery is part of the problem, I think the content is still a problem. I think your story is a perfect example – the content we “deliver” had no meaning to you. Much of the content in the Common Core has no meaning to the majority of our kids, and simply changing the delivery method is unlikely to fix that.

    While I agree that if we did a better job with our delivery there might be more students interested in math and science, even if we double that 3.9% we still are looking at over 90% of our students who have little need to be subjected to complex numbers (among other things).

  6. monika - I think you know I’m intrigued by what you’re trying, but I still have trouble seeing what that looks like in any kind of systemic way. Which is why I wonder whether it can be systematized.

  7. Pete Welter - I agree, yet I still have trouble figuring out what that looks like. How do you help students find their own questions that have mathematical content associated with them in any kind of systemic way? I want students to pursue their own passions, but I worry that we’ll eliminate the pursuit of passions that require a certain level of prerequisite knowledge to even know they might be passionate about something.

  8. Quick comments:

    Algebraic thinking finds authentic contexts for learning by examining real world environmental issues.

    Datasets abound on the web (e.g. My NASA Data) that can be used to examine the issue of Global Climate Change.

    Well-led student discussions of the meaning and the quality of the data supporting correlations between concentrations of "global warming" gases and the various measures of global temperatures all require mathematical reasoning that supports critical thinking on issues of global consequence.

    The devil is in the details of the discursive pedagogical approach. A master teacher gets students to talk and think in a holistic way that is difficult if not impossible to capture in bite-sized standards.

  9. How do you decide which students "need" algebra and which students don't? If you allow students to make that decision, you are allowing a 14-year-old to close off certain careers simply because math is not a passion of hers. I believe that is also the problem with personalized learning - we are assuming that an adolescent has the motivation and desire to work through a set of standards. Adolescents should not be expected to do that, nor should they be expected to have already decided what skills they will need for a future career.

  10. Tamarah - That's the dilemma. But do they all need a class called "Algebra," or can we find a better way to expose them to algebraic thinking that is more personalized?

  11. All algebra does is enable one to solve for the unknown. As long as there isn't any demand for the unknown, there won't be a demand for the ability to solve for it.
    Kids, along with the rest of us, are learning that the truth can't be substituted with the consumption of something else. This is why many of them are refusing to get married and buy houses. They move in back home with their parents because they've bought the dreams that were sold to them when they were young, but reject the impossibility of seeing them materialize, e.g. the clinic she wants to open up in Africa, or the design firm he wants to open up in Toronto.
    The will to take risks comes from having information... the unknown. This generation won't stop until they get what they came here for: living out their perception of the truth. And not having access to exactly this is all the incentive they need to solve for it. Just watch.

  12. I do believe Algebra is important to teaching children important critical thinking skills. I agree with a previous commenter that the content isn't the issue but the delivery. Working for an online educational company, Thinkwell, I've seen the difference an incredible teacher makes. Students go from being negative about math to not only loving the subject but the teacher as well.
    Many teachers today don't have the passion for math that gets kids excited about learning.
    I for one think some of this is a result of over testing. What do kids dread the most at school? Tests. So by forcing so much standardized testing on them, we are taking the passion and fun out of learning.
    Instead, we need to focus on sharing the joy of learning including different aspects that might not be considered as "educational" and on track like fractal geometry. Have it relate to their lives so children see learning benefits them and is a life-long adventure.